Currency Hedging for Long Term Investors with Liabilities

Transcription

1 Currency Hedging for Long Term Investors with Liabilities Gerrit Pieter van Nes B.Sc. April 2009 Supervisors Dr. Kees Bouwman Dr. Henk Hoek Drs. Loranne van Lieshout

2 Table of Contents LIST OF FIGURES... iv LIST OF TABLES... v LIST OF APPENDICES... vi Acknowledgements... vii Abstract Introduction Mean-Variance analysis Portfolio returns with currencies Funding ratio returns with currencies Mean-variance Optimization Data description and model specification Model Data and summary statistics Estimation results Dutch series Swiss series Term structure of assets and liabilities Correlation with asset returns Dutch case Swiss case Long duration bonds Correlation with price inflation and liability returns Optimal currency demands for investors with liabilities Base scenario Duration of the liabilities Duration matching Asset allocation Different percentage of bonds and equity Different exposures to foreign currency Indexation ambition Sub period Parameter uncertainty ii

3 6 Conclusion...50 References...52 Appendices...54 iii

4 LIST OF FIGURES Figure 1: Term structures of asset returns hedged with respect to the Euro and unexpected exchange rate returns Figure 2: Term structures of asset returns hedged with respect to the Swiss Franc and unexpected exchange rate returns Figure 3: Term structures of long duration bonds and unexpected exchange rate returns Figure 4: Term structures of price inflation and liability returns and unexpected exchange rate returns Figure A.1: Term structures of asset returns hedged with respect to the Euro and unexpected exchange rate returns for the sub period Figure A.2: Term structures of asset returns hedged with respect to the Swiss Franc and unexpected exchange rate returns for the sub period Figure A.3: Term structures of liability returns and unexpected exchange rate returns for the sub period iv

5 LIST OF TABLES Table 1: Data series and source Table 2: Summary statistics Table 3: Parameter estimates of the VAR-model for the Dutch case Table 4: Parameter estimates of the VAR-model for the Swiss case Table 5: Currency demands for the base scenario Table 6: Currency demands for different durations of the liabilities Table 7: Currency demands for different durations of the bond portfolio Table 8: Currency demands for different percentage bond and equity Table 9: Currency demands for different exposures to foreign currency Table 10: Currency demands for different indexation ambitions Table 11: Currency demands for the sub period v

6 LIST OF APPENDICES Appendix A.1: Derivation of the optimal mean-variance currency demands Appendix A.2: Covariance matrix for different investment horizons Appendix A.3: Term structures for the sub period vi

7 Acknowledgements I would like to make use of this opportunity to thank everyone involved in the completion of this thesis. First of all I would like to thank my supervisors, Kees Bouwman from the Erasmus School of Economics and Henk Hoek and Loranne van Lieshout from ORTEC Finance. All of you had very useful comments that helped to improve this thesis. Next I would like to thank my family and especially my parents who always have been supportive during the time of my studies and also provided financially. Thank you for who you are. Furthermore I would like to thank all of my friends who were always very interested in the process and my progress and also were very supportive. Finally I would like to thank my colleagues at ORTEC Finance for the nice working environment they provided and I m sure that it will continue to be great working with you now I will start working fulltime at ORTEC Finance. vii

8 Abstract This thesis considers a long term investor with liabilities that invests internationally and therefore runs currency risk and has to decide on how much of this risk it wants to hedge. The purpose of this thesis is to optimize this currency hedging decision. To achieve this, a meanvariance framework is derived that includes exchange rates and liabilities. It will be found that the investment horizon of the investor will play an important role in the optimal currency decision and that this decision also depends on the characteristics of the investor. Furthermore, it will be found that the total exposure to foreign currency in the portfolio is not important in determining the optimal currency decision but that this decision is mainly determined by the correlations of the exchange rates with the assets and the liabilities

9 1 Introduction Long term investors like pension funds usually hold internationally diversified portfolios. Among others, Solnik (1974) shows that investors can reduce total portfolio risk by including foreign securities, rather than diversifying their portfolios domestically. By holding foreign securities, an investor faces currency risk. The traditional view is that investors should fully hedge their currency exposure, because from the perspective of longrun policy, currency exposure should be seen as having zero expected return. Hedging therefore does not lower total expected portfolio return but it does reduce the risk and can therefore be seen as a free lunch as it is called by Perold and Schulman (1988). This point is also made by Jorion (1989), Kaplanis and Schaefer (1991) and Glen and Jorion (1992). Froot (1993) however argues that this is a short term argument and that it generally applies only if real exchange rates follow a random walk. He argues that when real exchange rates and asset prices display mean reversion, the optimal hedging decision of an investor will generally depend on the investment horizon. He shows that for longer horizons unhedged assets are less volatile then hedged assets. Because a pension fund is an investor with a relatively long investment horizon, it may therefore prefer to hedge less than fully. A drawback is that Froot focuses on how hedging effects the variance of an individual asset, where hedge ratios should be determined according to the effects on the entire portfolio. Campbell et al. (2003) give another argument for holding foreign currency. They argue that short term debt, which is usually seen as a riskless asset in a mean-variance framework isn t riskless in real terms in the long term 1. In the short term, the only risk arises from shocks to the price level, which are modest over short periods. In the long term it is no longer riskless, because the real interest rate varies over time and a long-term investor must roll over short term debt at uncertain future real interest rates. This risk can be hedged by holding foreign currency if the domestic currency tends to depreciate when the domestic real interest rate falls. Campbell et al. (2007) show that currency positions can be effective in managing the risk of 1 In fact, it is also not riskless in nominal terms

10 the total portfolio due to correlations between currency returns and equity returns. They consider seven countries and find that for some countries the currency exposure should be (partly) hedged or even over hedged and for other countries that the currency exposure should even be increased. The literature mentioned above considers the currency hedging policy for asset-only investors. Pension funds have for a long time been considered as asset only investors, because regulatory frameworks and accounting standards did not require fair valuation of pension liabilities. Recently, there has been a shift to fair valuation of liabilities and these should therefore also be considered in determining the optimal portfolio choices, since liabilities are also subject to risk like inflation risk and interest rate risk. Considering assets and liabilities together in a total balance sheet approach is called asset and liability management (ALM). Leibowitz (1987) and Sharpe and Tint (1990) determine the pension asset allocation through surplus management, thereby considering the pension liabilities. Sundaresen and Zapatero (1997) provide a framework in which they link the valuation and asset allocation policies of a pension plan with the lifetime marginal productivity schedule of the workers in the firm. Van Binsbergen en Brandt (2006) study the impact of regulations on the investment decisions of a pension plan by explicitly modeling the tradeoff between the long-term objectives and the short-term constraints. Hoevenaars et al. (2007) study the added value of alternative asset classes in an ALM-framework by maximizing the funding ratio return, a concept introduced by Leibowitz et al. (1994). The contribution of this thesis lies in combining the fields of currency hedging and asset and liability management. Current literature on currency hedging has not yet considered investors with liabilities and current literature on asset and liability management has not yet looked at exchange rate risk. The main purpose of this thesis will be to determine the optimal currency policy for an investor with liabilities for different horizons, given the characteristics of the investor and the asset allocation of the investor. Furthermore this thesis aims to give insight in how this optimal currency policy changes when the characteristics of the investor change or when the asset allocation of the investor changes. The insights obtained from this thesis will therefore be relevant for pension funds and other investors with liabilities like insurance companies, because it will be helpful in reducing the total balance sheet risk

11 The perspective of this thesis will be that of a Dutch and a Swiss pension fund. The pension systems in those countries have a lot of similarities. Moreover, the Swiss perspective is interesting because of the development of the Swiss Franc, which has not moved according to theoretical models. Furthermore, the US market and the British market will be used as foreign investment markets. A useful tool in studying the variances and covariances of different time series is the term structure of the risk-return tradeoff, introduced by Campbell and Viceira (2005). They estimate the covariance-matrix of a VAR (1) model for different horizons to see how the variances and covariances change with investment horizon. This method will be used in this thesis to model the variances and covariances for different horizons and these will be used in an optimization framework to determine the optimal currency policy. The set up of this thesis will be as follows. In section 2, an optimization framework will be set up. First the portfolio return will be derived for a hedged and unhedged portfolio. Next, the portfolio will be expanded with liabilities and at the end of that section an analytical expression will be derived for the optimal currency policy. Section 3 first discusses the VAR(1) model. Next a description of the data and of the applied data transformations will be presented as well as summary statistics of the data. The last subsection presents the parameter estimates for the VAR(1) model. Section 4 studies the term structure of the risk-return tradeoff. This will be done by looking at the correlation between the exchange rate returns and the different asset returns, liability returns and price inflation for different investment horizons. Section 5 presents the empirical results. First the optimal currency policy for a specified base scenario will be determined. It will be found that contrary to the findings of Froot (1993) for asset-only investors, investors with liabilities will hedge their currency positions and will even over hedge their positions for all investment horizons. Next a number of sensitivity analyses will be performed on the characteristics of the investor and on its asset allocation. The primary finding will be that the exposure in the portfolio to foreign currency does have little influence on the currency positions but that these are primarily determined by the correlations with the asset and liability returns. Section 6 concludes

12 2 Mean-Variance analysis This thesis considers the problem of a pension fund that invests in domestic and foreign bonds and stocks and must decide how much of the currency exposure it wants to hedge. This problem best reflects the situation of a pension fund that in general first determines the strategic asset allocation and then decides how much currency exposure it wants to run. The exposure to foreign currencies can be adjusted by entering into forward exchange rate contracts. First the portfolio return will be defined where the return definition of Campbell et al. (2007) will be followed. Because a pension fund also faces liabilities, these need to be taken into account. Therefore the framework will be expanded with liabilities following Hoevenaars et al. (2008). The framework will also be expanded to a multi-period framework. Finally the mean-variance optimization will be defined. 2.1 Portfolio returns with currencies In this section the portfolio return will be defined along the lines of Campbell et al. (2007). Let,, denote the gross return on asset in currency from holding asset for one period from time to time + 1, where asset can be domestic or foreign bonds or stocks. Let, denote the spot exchange rate in domestic currency per unit of foreign currency at time. The domestic country is indexed by = 1 and of course, the domestic exchange rate is constant over time and equal to 1 so, = 1 for all. An investor exchanges at time one unit of domestic currency for 1, units of foreign currency in the spot market and invests the proceeds in the bond or stock market of country to earn a return of,,. These returns can be exchanged at a rate of, so the unhedged returns can be written as,,,,. When the returns are stacked in the vector which denotes the gross returns in local currency, the unhedged portfolio return can be written as:, =, (2.1) where is a diagonal matrix with the weights of the assets in the portfolio,,,,,, on the diagonal. These portfolio weights always add up to one. The portfolio is build such that the first two elements correspond to domestic bonds and stocks

13 The following two elements correspond to bonds and stocks in the first foreign currency and so on. The number of currencies under consideration, including domestic currency, is therefore half the number of assets in the portfolio: =. The portfolio weights always add up to 1. Pension funds usually only have long positions and therefore it is assumed that the portfolio weights are positive. is the 1 vector of gross returns in local currency. D is a distribution matrix which has in each row at position a 1 when the asset is denoted in currency and 0 otherwise. In the case of three currencies under consideration and given the structure of the portfolio as described above, this distribution matrix would be given by: = The 1 vector has as its elements the spot exchange rates and denotes the elementby-element ratio operator so that when the i-th element of the vector corresponds to currency, then it is given by,,. To hedge part of the currency exposure, the pension fund can enter into forward exchange rate contracts. Let, denote the one-period forward exchange rate in domestic currency per unit of foreign currency and let, denote the domestic currency value of the amount of forward exchange rate contracts for the investments denoted in currency the pension fund enters into at time per unit of domestic currency invested in the portfolio. Because, is the value per unit invested in the portfolio,, will be referred to as the hedge position. At time + 1 the pension fund can exchange,, of the returns denoted in currency,,,, at an exchange rate of,. The remaining part,,,,,,, will be exchanged at the spot exchange rate,. The vector with hedge positions will have a length that is half that of the number of assets in the portfolio. Putting everything together the partly hedged portfolio return can be written as:, = +, (2.2) - 6 -

14 where is the 1 vector of forward exchange rates, and =,,,,,,. Because, =, = 1 for all, the value of the hedge position of the domestic assets is arbitrary. Therefore it is set such that the sum of all hedge positions is 1. Therefore it holds that:, = 1, (2.3) When covered interest parity holds, the forward contract for currency trades at, =, 1 +, 1 +,, where, denotes the domestic short-term interest rate and, is the short-term interest rate of country. When this expression for the forward contract is substituted in equation (2.2), the hedged portfolio return can be written as:, = + + +, (2.4) where is the 1 vector of ones, is the 1 vector with foreign short-term interest rates and is the 1 vector with as each element the domestic short-term interest rate. From equation (2.4) it can be seen that selling currency forward is analogous to a strategy of going short in foreign cash and invest the proceeds in domestic cash or selling foreign currency and lending domestically. The pension fund is said to have fully hedged its currency exposure when it sets the hedge position, equal to the weights of the investments denoted in currency. This is given by, +,. It under-hedges its currency exposure when it sets, <, +, and it over-hedges its currency exposure when it sets,, >, +,. When the pension fund chooses to fully hedge, it must be noted that the position is not exactly hedged, because the position fluctuates with realized return. By using forward contracts, a pension fund can choose how much currency exposure it wants to run. Therefore, the portfolio return can also be written in terms of exposure instead of hedge positions. To this end, a new variable is introduced:,, +,,,, so when the pension fund does not want to have any exposure at all, it sets, = 0. A positive value of, means that the pension fund is not fully hedging its position and that it wants to have a currency exposure or equivalently, it has a demand for currency. Now equation (2.4) can be written in terms of currency demands: - 7 -

15 , = , (2.5) where =, +,,, +,,,, +, and =,,,,,,. It follows that =. Because the portfolio weights add up to one, equation (2.3) implies:, =,, (2.6) or =, so that, represents domestic currency exposure. It is also easily seen that the currency portfolio is a zero investment portfolio. Since the portfolio weights add up to one, the portfolio is fully invested in the assets. Therefore, the only way to achieve an exposure to a currency is to go short in another currency and the result is a zero investment portfolio. It is easier to work with log returns. Therefore, a log version of equation (2.5) is needed. Campbell et al. (2007) derive this log version and show that it is approximately equal to:, = , (2.7) where the third term is a Jensen s variance correction term and is equal to = , (2.8) where the operator stacks the diagonal elements of a matrix into a vector. For ease of notation the following vectors will be defined: + + The intuition behind these variables is as follows. is simply the vector of hedged returns of the different asset classes. Exchange rate returns can be divided in an expected return which is explained by the interest rate difference and an unexpected return. is the vector with as elements the unexpected part of the exchange rate returns. Substituting these variables in equation (2.7) and (2.8) yields: - 8 -

16 , = + + (2.9) = (2.10) 2.2 Funding ratio returns with currencies Because a pension fund also faces liabilities, these need to be taken into account. Therefore the framework will be expanded with liabilities following Hoevenaars et al. (2007). They approach asset-liability management from a funding ratio perspective, a concept first introduced by Leibowitz et al. (1994). The advantage of working with the funding ratio return is that it is independent of the initial funding ratio. The funding ratio (F) is defined as the assets (A) of a pension fund divided by its liabilities (L). The funding ratio log-return is then defined as the log return of the assets minus the log return of the liabilities:, =,, (2.11) This expression can also be interpreted as the relative change in the funding ratio. The log return on the assets is defined in equation (2.9). Substituting in equation (2.11) yields:, = + +, (2.12) When the portfolio weights are kept constant over different periods, the multi-period funding ratio return can be obtained by simply adding the single-period funding ratio returns. This is what is also seen in practice for pension funds. They usually perform an ALM study to determine the strategic asset allocation for several years and once this has been determined, the portfolio will be rebalanced to the strategic weights on a regular base. Therefore the multiperiod funding ratio return is given by:, =, =, +, +, (2.13) The subscript + denotes the cumulative return periods from period to +. The vector contains the horizon dependent currency exposures. To come to the mean and variance - 9 -

17 of the multi-period funding ratio returns, first the annualized expected returns and the annualized covariance matrix will be defined. Therefore the returns are stacked in the vector: = (2.14), The annualized expected returns and the annualized covariance matrix are now given by: = E =, (2.15) = Var = σ σ σ σ σ (2.16) The notation denotes the cumulative excess return in the period from to +. The covariances change as the investment horizon changes. This relation between the investment horizon and the annualized covariance matrix is the term structure of the risk-return tradeoff which is introduced by Campbell and Viceira (2005). The final step is evaluating the mean and variance of the multi-period funding ratio return: E, = + +, (2.17) Var, = ( + + σ σ 2 σ ) (2.18)

18 2.3 Mean-variance Optimization The purpose is to determine the optimal currency policy for a long term investor with liabilities given the portfolio and the liabilities. Given the definitions before, this comes down to determining the optimal currency exposure. Therefore it is assumed that the portfolio weights are given and that the choice variable is, the vector with as its elements the horizon dependent currency demands. Note that, need not to be determined, because it corresponds to the investments in the domestic currency in the portfolio. As can be seen from equation (6), its weight is given once the other currency demands are determined. Therefore this weight is excluded from the choice variable and the adjusted vector is now given by: =,,,, Following Van Binsbergen and Brandt (2006) it is assumed that a pension fund has constant relative risk aversion (CRRA) preferences on the funding ratio at some future date = + : = max {,, }, where λ 0. (2.19) This is a standard power utility function, where the parameter λ can range from zero to infinity. This parameter λ is typically interpreted as a measure of the risk tolerance of an investor. Pension funds will in general be risk averse, because they are bound by certain risk constraints by regulatory authorities. Therefore in this thesis it is assumed that λ > 1. The difference with Van Binsbergen and Brandt is that here it is assumed that the portfolio weights are fixed over the investment horizon. Therefore the problem to be solved is similar to that of Hoevenaars et al. (2008). They state that when normality of the excess returns is assumed, the optimization problem of equation (2.19) reduces to: max 1 λvar, +, (2.20) After some algebraic manipulation shown in the appendix, this problem leads to the following vector of optimal mean-variance currency demands:

19 = λ σ + + (2.21) When looking at the term in the second squared brackets, it can be seen that it can be split in a part that is multiplied by 1 λ and a part that is not. The part that is multiplied by 1 λ depends entirely on the horizon dependent covariances between the unexpected exchange rate returns and the asset and liability returns. This part can therefore be seen as the hedge demand and will by itself result in the currency positions that, given the portfolio weights, will minimize the variance of the funding ratio return. The other part also concerns returns and variances and can be seen as a speculative portfolio. This speculative portfolio is therefore given by: = 1, + + (2.22) The hedge demand is given by: = σ (2.23) When a pension fund would be extremely risk averse, it will let λ go to infinity and it is easily seen that in that case the speculative portfolio will be zero and the only part left is the hedging demand. When looking at the expression in equation (2.23) it can be seen that when the covariance between the unexpected exchange rate returns and the liability return increases, that also the demand for currency increases. The term in squared brackets will not change when this covariance changes. Because it is assumed that λ > 1 the term 1 will have a negative sign and that causes the covariance matrix between the unexpected exchange rate returns and the liability return to have a positive sign. In this case that means that when the liabilities increase, the unexpected exchange rate return tends in the same direction and in that way they have an offsetting effect on the funding ratio. When this tendency becomes stronger, the demand for currency will increase. Along these lines it can be said that when the covariance between the unexpected exchange rate returns and the liability returns has a

20 positive sign, it will contribute to a positive demand for currency and when it has a negative sign, it will contribute to a negative demand or stated otherwise, an over hedged currency position. By the same reasoning does the horizon dependent covariance matrix between the unexpected exchange rate returns and the hedged asset returns have a negative sign. When the covariance between the unexpected exchange rate returns and the bond and stock returns decreases, the demand for currency will increase. When the covariances decrease, there will be better diversification possibilities, which increases the demand for currency. Also when the sign of the covariance is positive, it will contribute to a negative demand for currency or an over hedged currency position and when the sign of the covariance is negative, it will contribute to a positive demand for currency. Equation (2.22) also contains the covariance matrix between the unexpected exchange rate returns and the hedged asset returns but this time it is the one year horizon covariance matrix. When the pension fund will not be that risk averse equation (2.22) will also partly attribute to the currency demand and therefore the effect described above will become stronger. From this equation it can also be seen that when the unexpected exchange rate return increases, the demand for currency will increase and that when the variance of the exchange rate returns decreases, the demand for currency will increase

21 3 Data description and model specification This section describes the model that will be used for the modeling of the return dynamics and describes the data that will be used. The model that will be used is a vector-autoregressive (VAR) model and will be described in section 3.1. The data to be used as well as summary statistics of the data will be discussed in section 3.2. This section ends with the estimation results in section Model From section 2.2 it became clear that the interest lies in the covariances between the returns on assets and liabilities and the unexpected exchange rate returns for different investment horizons. This covariance structure can be derived by constructing a VAR model of order one. A VAR model is a relatively simple model in which current values of economic variables and asset- and liability returns are linearly related to past values of the same set of variables. In mathematical notation the VAR model of order one is given by: = +, ~0, (3.1) where is the vector with returns as specified in equation (2.14) and is a vector of residuals or one-step forecast errors which are assumed to be multivariate normally distributed with a zero mean vector and covariance matrix. This covariance matrix is the matrix as specified in equation (2.16). The data that are used are year data, so the covariance matrix corresponds with an investment horizon of one year. It can be computed for different investment horizons using the estimated VAR coefficient matrix. This coefficient matrix describes the linear relation between current and past values of the returns. Although the VAR-model has a relatively simple structure, it is very well able to describe the most important dynamic characteristics of the annually observed returns. These characteristics do not only include the averages and standard deviations of the returns, but also the correlations and auto and cross correlations

22 3.2 Data and summary statistics The perspective of this thesis is that of a Dutch and a Swiss pension fund, investing domestically and also in the United Kingdom and in the United States and are therefore facing currency risk with respect to the British Pound and the US Dollar. For the Dutch case, investing domestically means investing in fixed income in the Netherlands and investing in equity in a European index. Therefore the analysis will be based on the short- and long term interest rates and equity returns in these countries, the exchange rates between these countries and the price inflation in the Netherlands and Switzerland. The price inflation is needed since pension liabilities are (conditionally) indexed with price inflation, depending on the financial situation of a pension fund. This conditionality is expressed as a function of the funding ratio of a pension fund. In practice, some pension funds index their active participants with wage inflation but here it is assumed that both active and inactive participants are indexed with price inflation. An overview of the data that will be used, as well as their source can be found in table 1. All data are end of year data from 1970 to 2007 unless stated otherwise. First logs of the series are taken. The equity series are already return series so these series need no further transformation. They are turned into hedged series by adding the logarithm of the domestic short term interest rate and subtracting the logarithm of the foreign short term interest rate. The log bond returns are calculated from the long nominal interest rate series by using the approximation of Campbell et al. (1997) which is given by:, =,,, (3.2) where n is the maturity of the bond and, is the log long nominal interest rate: ln1 +, at time., will be approximated by,. These series are also turned into hedged series in the same way as described before. is the duration of the bond. In this thesis, the duration will be an input parameter and will therefore not depend on time and on maturity. The 10 year interest rate will be used for the long interest rate

23 Region Netherlands Switzerland Panel A: Price Inflation Source Bloomberg: OENLC005 Bloomberg: OECHC006 Region Netherlands Switzerland United Kingdom United States Region Netherlands Switzerland United Kingdom United States Region Netherlands Switzerland United Kingdom United States Panel B: Short nominal interest rate Panel C: Long nominal interest rate Panel D: Equity returns Source : van de Poll (1996) : Bloomberg: NEC0YL03 Bloomberg: OECHR : Bank of England, 1 year nominal rate : Bloomberg: GUKTB3MO Bloomberg: USGG3M Source Bloomberg: OENLR006 Bloomberg: OECHR006 Bloomberg: OEGBR006 Bloomberg: USGG10YR Source : MSCI 3 Europe Gross Index local : MSCI Europe Gross Index Euro MSCI Switzerland Gross Index local MSCI United Kingdom Gross Index local MSCI North America Gross Index - USD Panel E: Exchange rates (in Euro) Region Source Swiss Franc 4 (CHF) Bloomberg: OECHK003 British Pound (GBP) Bloomberg: OEGBK004 US Dollar (USD) Bloomberg: OENLK002 (inverse) Table 1: Used data series, end of year data from unless stated otherwise and their source. For a long time, liabilities have been discounted with a fixed interest rate and would not fluctuate with changes in the interest rate and were therefore not subject to interest rate risk. Recently, there has been a shift to fair valuation of the liabilities and these will therefore change when a change in the discount rate occurs. The duration of liabilities is in general much larger than the duration of the bond portfolio of a pension fund. The pension fund therefore has a duration mismatch. A lot of pension funds apply the principle of duration 2 Van de Poll (1996), Bronbeschrijving gegevens voor onderzoek risicopremie Nederlandse aandelen, Onderzoeksrapport WO&E nr 465/9615, DNB The Bloomberg series for CHF and GBP are expressed in USD and have been converted to Euro values using the Euro/USD series

24 matching where the duration of the bond portfolio is increased by adding long maturity bonds to the portfolio or by adding interest rate swaps. By bringing the duration of the bond portfolio more in line with the duration of the liabilities, the pension fund will be less sensitive to changes in the interest rate. An interesting question therefore will be if a pension fund who has matched the duration of the bond portfolio with the duration of the liabilities will have another demand for currency. The exchange rate series are calculated using the following steps. First the exchange rate returns are calculated using, = ln, ln,. Then the unexpected part of the exchange rate return is calculated as:, =, + ln1 +, ln1 +, (3.3) The liability returns are also constructed by the approximation of Campbell et al. (1997) but also an inflation term is added to account for the (conditional) indexation of the liabilities. Here it is assumed that indexation is granted conditionally and therefore the liabilities have to be discounted by the nominal interest rate. The liability returns are therefore constructed in the following way:, =,,, + (3.4) where is the log price inflation and is the indexation ambition of the pension fund. Since indexation is granted conditionally, indexation will in general not be granted fully at all times and therefore an ambition is specified of the percentage of indexation a pension fund want to reach over a certain period. Most pension funds set their ambition equal to around 80%. The assumption of conditional indexation best reflects a Dutch pension fund, because the majority of Dutch pension funds are granting indexation conditionally. Swiss pension funds however grant indexation unconditional. For sake of comparability it is assumed that a Swiss pension fund will also grant indexation conditionally. In one of the analysis, there will also be looked at a situation where unconditional indexation is the case

25 There are some conditions to assume that the liabilities of a pension fund can be described as a constant maturity (indexed-linked) bond. Hoevenaars et al. (2008) state that a sufficient condition for this to be true is that the distribution of the age cohorts and the accrued pension rights are constant through time and that the inflow from contributions are equal to the net present value of the new liabilities. For the base scenario it is assumed that the pension fund has no active duration matching policy and has a duration of its bond portfolio of 5. The duration of the liabilities is set equal to 20. The indexation ambition is set equal to 80%. There will also be looked at the currency demand for pension funds that have matched the duration of the assets and the liabilities and for pension funds that have a higher or lower duration of their liabilities. The series for the hedged bond and stock returns, the unexpected exchange rate returns and the liability returns as described in equations (3.2) (3.4) are constructed using the input parameters of the base scenario. Summary statistics of these series containing data from can be found in table 2. Also presented are the summary statistics for the two sub periods and The first period is known for its high inflation and interest rate levels. Also at the beginning of the 1980 s the central banks changed their monetary policy to using the interest rates as policy instruments for controlling inflation. Also shown are liability returns in case that pensions would not be indexed. The average return on bonds in the Netherlands and on UK and US bonds, hedged with respect to the Euro is around 7%. Swiss bonds and UK and US bonds, hedged with respect to the Swiss Franc have an average return of around 5%-5,5%. The standard deviation of the returns on bonds in the Netherlands and in Switzerland is 4,43% and 3,84% which is lower than the standard deviation of bond returns in the UK and US, irrespectively to which currency they have been hedged and range from 6%-8%. When looking at the different sub periods it can be seen that except for US bonds, the returns are higher in the first period. Also the standard deviation for all bonds is higher in the first sub period. The average stock returns are also higher for the Netherlands and the UK and US hedged with respect to the Euro than for Switzerland and the UK and US hedged with respect to the Swiss Franc. The first range from 9,6% to 11,1% and the second range from 7,9% to 9,2%

26 Period Mean Stdev Mean Stdev Mean Stdev Hedged bond Returns Netherlands 7,09% 4,43% 8,25% 4,73% 6,53% 4,26% Switzerland 4,47% 3,84% 5,54% 4,42% 3,96% 3,51% United Kingdom (EUR) 6,89% 7,90% 8,34% 12,36% 6,19% 4,70% United States (EUR) 7,22% 6,01% 6,74% 7,25% 7,45% 5,47% United Kingdom (CHF) 5,21% 7,54% 6,27% 11,69% 4,70% 4,66% United States (CHF) 5,54% 6,44% 4,67% 7,86% 5,96% 5,77% Hedged Stock Returns Netherlands 11,14% 18,82% 8,69% 19,35% 12,31% 18,85% Switzerland 9,19% 21,31% 3,54% 19,35% 11,90% 22,04% United Kingdom (EUR) 9,56% 16,92% 9,20% 22,95% 9,74% 13,71% United States (EUR) 10,46% 15,90% 7,58% 19,04% 11,85% 14,38% United Kingdom (CHF) 7,89% 16,21% 7,13% 21,43% 8,25% 13,54% United States (CHF) 8,78% 15,56% 5,51% 18,04% 10,36% 14,35% Unexp Exchange Rate Returns GBP-EUR -0,37% 10,29% -3,22% 11,91% 1,01% 9,37% USD-EUR -2,26% 12,71% -2,52% 10,87% -2,14% 13,72% GBP-CHF 0,03% 11,78% -4,96% 14,07% 2,43% 9,94% USD-CHF -1,86% 13,51% -4,26% 13,23% -0,71% 13,75% Liability Returns Netherlands 11,25% 17,76% 12,98% 19,72% 10,42% 17,11% Netherlands (not indexed) 8,40% 17,82% 7,52% 20,03% 8,82% 17,08% Switzerland 7,76% 15,45% 11,76% 17,47% 5,84% 14,36% Switzerland (not indexed) 5,48% 15,84% 7,82% 18,82% 4,36% 14,50% Table 2: Summary statistics for the different series containing yearly data from and for the sub periods and , constructed under the assumptions of the base scenario. For the hedged series is stated in parentheses with respect to which currency the series is hedged. The standard deviation of the stock returns, contrary to bond returns, is higher in the Netherlands and in Switzerland than in the UK and US, irrespectively to which currency they have been hedged. For the Netherlands and Switzerland it is equal to 18,82% and 21,31% and for the UK and US it ranges from 15,5% to 17%. What is remarkable is that for the second sub period, compared to the first sub period, all the average hedged stock returns are higher and that except for Switzerland, all the standard deviation are lower. For US and UK equity, irrespectively to which currency they have been hedged it is even lower by 4 to 8 percentage point. The average unexpected exchange rate returns are negative for the GBP-EUR, USD-EUR and USD-CHF series and is approximately zero for the GBP-CHF series. In these currency pairs, the former is the foreign currency and the latter is the domestic currency. The USD series have an average unexpected exchange rate return close to -2% and the average return for the GBP-EUR series is equal to -0,37%. The standard deviation of these series ranges from 10,3%

27 to 13,5%. Therefore none of the unexpected exchange rate series has a mean significantly different from zero. When looking at the sub periods it can be seen that only the USD-EUR series has approximately the same mean for the two sub periods. The mean for the British Pound series even changes sign and has a difference of 4% with respect to the Euro and even 7% with respect to the Swiss Franc. The average liability return in the Netherlands is equal to 11,25% and has a standard deviation of 17,76%. The average liability return in Switzerland equals 7,76% and has a standard deviation of 15,45%. This return is higher for the Netherlands because the long nominal interest rate has been higher in the Netherlands over the sample period and also the price inflation has been higher. These liability returns show that a pension must hold a portfolio of bonds and stocks to accomplish a higher average expected return on the assets than the return on the liabilities. It can also be seen by looking at the average returns for the full period that for a Dutch pension fund an indexation ambition of 80% and at the same time keeping the funding ratio stable cannot be realized unless extra contributions are made because the indexation cannot be financed by the excess return. None of the asset classes has an average return that is at least equal to the average return on the liabilities. When looking at the sub periods it can be seen that an ambition of 80% can be reached in the second period due to lower price inflation and in the Swiss case also due to lower interest rates. The standard deviation is 2.5 to 4 percentage point lower in the second sub period compared to the first sub period. 3.3 Estimation results This section reports the parameter estimates of the VAR model as described in equation (3.1) as well as the correlation matrix of the residuals. Section reports the estimates for the Dutch series and section reports the estimates for the Swiss series

28 3.3.1 Dutch series The parameter estimates of the coefficient matrix A of the VAR model as described in equation (3.1) for the Dutch series, as well as the correlation matrix of the residuals can be found in table 3. Also stated are the t-statistics of the parameter estimates. Panel A: Parameter estimates of the coefficient matrix A of the VAR model,,,,,,,,,,,,,,,,, 1,92-0,07 0,07 0,01-0,04 0,00-0,13 0,10-0, (2,92) (-0,71) (0,39) (0,12) (-0,20) (0,02) (-1,38) (1,49) (-3,49),, 5,04-0,58-1,04 0,53-0,60 0,24 0,75-0,13-0, (1,67) (-1,33) (-1,30) (1,00) (-0,63) (0,67) (1,80) (-0,42) (-0,52),, 2,94-0,30-0,35 0,10 0,15 0,13-0,10 0,22-0, (2,52) (-1,77) (-1,13) (0,50) (0,42) (0,97) (-0,61) (1,82) (-2,46),, 4,68-0,48-0,51 0,27 0,08 0,30 0,43-0,02-0, (1,65) (-1,16) (-0,68) (0,54) (0,09) (0,92) (1,08) (-0,08) (-1,18),, 1,67-0,02-0,08 0,06 0,26-0,10-0,09 0,18-0, (1,77) (-0,12) (-0,31) (0,37) (0,87) (-0,95) (-0,66) (1,81) (-2,54),, 3,15-0,29-0,39 0,33 0,03 0,05 0,50-0,32-0, (1,12) (-0,71) (-0,53) (0,66) (0,03) (0,14) (1,27) (-1,07) (-0,64), 0,30-0,59-0,88 0,42-0,20 0,42 0,16 0,04 0, (0,18) (-2,45) (-2,02) (1,45) (-0,39) (2,18) (0,71) (0,21) (0,63), 2,69-0,31 0,37-0,30-1,03 0,62 0,29 0,24-0, (1,51) (-1,19) (0,79) (-0,96) (-1,84) (2,98) (1,18) (1,29) (-0,91), 3,97-0,32 0,30 0,06-0,17 0,02-0,55 0,47-1, (1,31) (-0,73) (0,38) (0,12) (-0,18) (0,06) (-1,31) (1,46) (-1,69) Panel B: Correlation matrix of the residuals,,,,,,, 1, 0,365 1, 0,581 0,451 1, 0,396 0,896 0,675 1, 0,730 0,400 0,460 0,332 1, 0,273 0,807 0,413 0,753 0, ,386-0,018-0,010 0,029-0,519 0, ,002 0,283 0,116 0,253-0,240 0,058 0, ,998 0,360 0,566 0,383 0,727 0,269-0,355 0,020 1 Table 3: Panel A contains the parameter estimates of the coefficient matrix A of the VAR model for the Dutch series. In parenthesis are the t-statistics of these estimates. The last column contains the regression. Panel B contains the correlation matrix of the residuals

29 The first thing that becomes clear from examining these statistics is that the majority of the parameter estimates are not significant and that the hedged stock return series and the liability return series have no significant parameter estimates at all. Still these estimates are useful for constructing the Term structures of risk which will be discussed in section 4. Also remarkable is the almost perfect correlation between the domestic bond returns and the liability returns. This can be explained by the fact that both series are constructed using the same underlying series, the long term interest rate. They only differ in duration and in the inflation part. What is also of interest is the conditional correlation of the unexpected exchange rate returns with the other returns. Desirable is a negative and preferably low correlation with the bond and stock return series because this will give rise to diversification possibilities and a positive and preferably high correlation with the liability return, because than the unexpected exchange rate returns serve as a hedge. These will be examined more closely in section 4 when also is looked at these correlations at different investment horizons Swiss series The parameter estimates of the VAR model as described in equation (3.1) for the Swiss series, as well as the correlation matrix of the residuals can be found in table 4. Also stated are the t- statistics of the parameter estimates. There are a lot of similarities when comparing these results to the results for the Dutch series as discussed in the previous section. Here also the majority of the parameter estimates is not significant and the hedged stock return series have no significant parameter estimates at all. The liability return series, contrary to the Dutch case, does have one significant parameter estimate. This is the domestic bond return series. Also in the Swiss case there is an almost perfect correlation between the domestic bond return and the liability return

30 Panel A: Parameter estimates of the coefficient matrix A of the VAR model,,,,,,,,,,,,,,,,, 3,92 0,01 0,10-0,02-0,12-0,03-0,07 0,02-0, (4,09) (0,12) (0,69) (-0,29) (-1,02) (-0,50) (-1,06) (0,49) (-4,51),, 10,50-0,50-1,30 0,62 0,57-0,16 0,89 0,27-1, (1,46) (-1,40) (-1,14) (1,07) (0,64) (-0,35) (1,73) (0,76) (-1,10),, 6,09-0,14-0,46 0,07 0,30-0,01-0,05 0,22-1, (2,88) (-1,38) (-1,36) (0,42) (1,15) (-0,05) (-0,32) (2,10) (-3,02),, 7,31-0,24-0,77 0,29 0,51-0,03 0,55 0,06-1, (1,30) (-0,85) (-0,86) (0,64) (0,72) (-0,08) (1,36) (0,24) (-1,10),, 5,29 0,01-0,10 0,08 0,21-0,16-0,03 0,15-1, (3,12) (0,14) (-0,37) (0,62) (1,00) (-1,53) (-0,26) (1,81) (-3,83),, 6,40-0,09-0,42 0,32 0,47-0,18 0,56-0,26-1, (1,15) (-0,33) (-0,47) (0,72) (0,68) (-0,51) (1,40) (-0,97) (-1,08), -7,06-0,44-0,82 0,30 0,33 0,37 0,05 0,08 1, (-1,97) (-2,46) (-1,43) (1,03) (0,74) (1,63) (0,19) (0,46) (2,47), -5,37-0,32 0,82-0,32-0,67 0,62-0,05 0,31 1, (-1,33) (-1,61) (1,27) (-0,97) (-1,32) (2,44) (-0,16) (1,56) (1,50), 13,57-0,01 0,35-0,04-0,67-0,12-0,32 0,10-3, (3,14) (-0,03) (0,51) (-0,10) (-1,23) (-0,44) (-1,05) (0,48) (-3,40) Panel B: Correlation matrix of the residuals,,,,,,, 1, 0,352 1, 0,638 0,383 1, 0,293 0,759 0,697 1, 0,539 0,434 0,282 0,206 1, 0,169 0,811 0,401 0,760 0, ,097 0,243 0,218 0,276-0,215 0, ,030 0,457 0,379 0,500 0,065 0,440 0, ,994 0,353 0,622 0,293 0,544 0,182-0,087 0,019 1 Table 4: Panel A contains the parameter estimates of the coefficient matrix A of the VAR model for the Swiss series. In parenthesis are the t-statistics of these estimates. The last column contains the regression. Panel B contains the correlation matrix of the residuals

31 4 Term structure of assets and liabilities From section 2 and 3 it became clear that of particular interest are the correlations of the unexpected exchange rate returns with the asset returns and the liability returns and therefore also with the price inflation since that is a part of the liability return. Even more of interest is how these correlations change with the investment horizon. A useful tool to study how the correlations change over time is the term structure of the risk return tradeoff which is introduced by Campbell and Viceira (2005). They estimate a VAR(1) model and construct the variance covariance matrix for different investment horizons as described in appendix A.2. This term structure gives the correlation between cumulative returns over different horizons. A drawback of this method is that short term data (year data) are used to estimate the longterm characteristics. However, since a dataset containing 200 year of data is not available, it is an acceptable method to work with. Section 4.1 studies the term structure of the unexpected exchange rate returns with the hedged asset returns. Section 4.2 studies the term structure of the unexpected exchange rate returns with the liability returns and price inflation. 4.1 Correlation with asset returns This section studies the term structure of the unexpected exchange rate returns with the hedged asset returns. This will be done for the base-scenario as defined in section 3.2. Also attention will be paid to the correlation of the unexpected exchange rate returns with long duration bonds. The duration will be set such that it equals the duration of the liabilities and is therefore equal to 20. Section will study the term structure for the Dutch case and section will study the term structure for the Swiss case. The long duration bonds will be studied in section Dutch case Figure 1 shows the term structures of the unexpected exchange rate returns with the hedged asset returns. The solid line represents the British Pound and the dashed line the US Dollar. When looking at the Dutch bonds it can be seen that the correlation with the British Pound decreases at first and after six years starts increasing again and stabilizes at its short term level again after 20 years. Diversification possibilities and therefore demand for currency will be